The two complexes considered above correspond to the ASD and flip' cases in [1]} and the index can be seen to match (upon applying (REF )). Finally, also in this case \(_{\mathrm {eq},H}(E^\bullet _2,\eth _2)\)  is symmetric under a reflection \((n_1,n_2)\rightarrow (-n_1,-n_2)\)  and the final result is independent of the initial choice of deformation map.
